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Aspects of Spatial Scaling

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We need to relate measurements at different scalesLabLogsCrosswellVSPSurface Seismic

How does laboratory rock physics apply to the field ?

frequency differences sample size differences wavelength differences

Seismic velocity depends not just on the rock andfluid properties, but also on the measurement scalerelative to the geologic scale

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Scale effects on measured velocity

Marion et al. (1994)

-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50

RT

Time (s)

EMT

J.1

Waves were propagated through periodic media created by

stacking plastic and steel disks. At the top, the effective

layer thickness is large compared with the wavelength; at

the bottom it is small compared with the wavelength. The

waveforms show that both the travel time andamplitude/frequency depend on the ratio wavelength to layer

thickness. The velocities in the two limits are described

wellby ray theory and effective medium theory, respectively.

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J.2

Apparent velocity picked from the layered medium

experiment (top) and numerical simulations of the

experiment (bottom).

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One dimensional scale effectsin layered media

Effective medium limit ( >> d):

Ray theory limit (

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Random arrangement of high and lowimpedance layers: a laboratory VSP

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Scale effects in a laboratory simulated VSP

20

40

60

80

100

120

140

0 4 8 12 16 20 24 28 32

experimental

5% pick (propagator matrix)

10% pick (propagator matrix)

20% pick (propagator matrix)

approximate recipe

Kennet-Frazer

numberofdisks

propagation times (microseconds)

effective medium

ray theory

=20=27

J.3

In a second experiment the plastic and steel disks were

stacked randomly, to create a medium with random plastic

and steel interval thicknesses. Waves were propagated

through the growing stack, to roughly simulate a VSP.

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Scale Effects on Seismic Velocities

Interval velocities in thinly layered media

J.4

This slide shows the apparent interval velocity in each

plastic interval in the laboratory VSP. The difference

of arrival times picked from the waveforms at the top

and bottom of each plastic interval were divided into

the interval thickness to get the velocity. The bulkplastic velocity is ~2500 m/s. We see that this ray

theory approach gives nonsense values when the

interval thickness becomes small relative to the

wavelength.

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Short wavelength and long wavelength syntheticseismograms for plane wave propagation througha 2-D random heterogeneous medium with Gaussianspatial autocorrelation function

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Velocity Dispersion in2-D Random Heterogeneous Media

3600

3620

3640

3660

3680

3700

3720

0.1 1 10 100

Velocity

/a

effective medium

ray theory 1-D

ray theory 2-D

ray theory 3-D

Comparison of numerical wave propagation in 2-Dheterogeneous medium with a Gaussian spatialautocorrelation function and ray theory predictionsof Boyse (1986)

J.5

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Short Wavelength Behavior

The variance of the traveltime fluctuations aroundthe mean traveltime can be related to the varianceof the slowness fluctuations (Mller et al., 1992)

For plane waves in a heterogeneous medium witha Gaussian spatial autocorrelation function:

L : pathlengtha : spatial correlation length

T2 : traveltime varianceS2 : slowness variance

T

2

= LaS

2

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Short Wavelength Behavior

Ray theory in random media

difference between ray theory slownessand mean slowness:

(Boyse, 1986)

Rss : spatial autocorrelation functiona : correlation length

S2 : slowness varianceL : pathlength

S= S0+ S

S= SRT

S0=

S

2 L

a

D

D = RSS

0

d

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0.6

0.7

0.8

0.9

1.0

1.1

0 20 40 60 80 100 120 140

Sandstone

Vrb/Vus

1/Q

Extension

Shear

Pressure

0.6

0.7

0.8

0.9

1.0

1.1

0 20 40 60 80 100 120 140

Limestone

Vrb/Vus

1/Q

Extension

Shear

Pressure

Velocity dispersion versus attenuation for sandstoneand limestone samples (Lucet, 1989)

J.6

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Velocity Dispersionin Heterogeneous Limestone

4900

4950

5000

5050

5100

5150

5200

5250

5300

0 0.1 1 10 100

velocity

/a

ray theory 3-D

ray theory 2-D

effective medium

Comparison of velocities computed from averagetraveltimes in numerical simulations with theoreticalpredictions

x-ray

image

J.7